1) Suppose that A is an nxn nonsingular upper triangualr matrix, and let B=A^-1. You may assume that all diagonal elements of A are nonzero.
a) using only the definition of matrix multiplication prove that Bsubnj =0 for all j<n. ( ie, show that all elements in the nth row of B, except for the last element, are zero.)
b) using only the definition of matrix multiplication , prove that B is upper triangular.

Thanks.

August 3rd 2008, 12:01 AM

CaptainBlack

Quote:

Originally Posted by happystudent

1) Suppose that A is an nxn nonsingular upper triangualr matrix, and let B=A^-1. You may assume that all diagonal elements of A are nonzero.
a) using only the definition of matrix multiplication prove that Bsubnj =0 for all j<n. ( ie, show that all elements in the nth row of B, except for the last element, are zero.)
b) using only the definition of matrix multiplication , prove that B is upper triangular.

Thanks.

Let the columns of be , then by definition .

Let the -th row of be .

Then the -th element in the -th row of is:

Now, as is upper trianglar:

and if this is zero and as by definition for

Now suppose that for some for all

Then:

because is upper triangular, and:

because by hypothesis , for all .

Hence:

So

Which completes a proof by induction that for all , which proves that is upper triangular.

(it is easier to see what is going on doing this element by element along the a row of )